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Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2*x + (1-4*k)*x^2)).
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%I #36 May 12 2021 03:54:53

%S 1,1,1,1,1,1,1,1,2,1,1,1,3,4,1,1,1,4,7,9,1,1,1,5,10,21,21,1,1,1,6,13,

%T 37,61,51,1,1,1,7,16,57,121,191,127,1,1,1,8,19,81,201,451,603,323,1,1,

%U 1,9,22,109,301,861,1639,1961,835,1

%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2*x + (1-4*k)*x^2)).

%H Seiichi Manyama, <a href="/A306684/b306684.txt">Antidiagonals n = 0..139, flattened</a>

%F A(n,k) is the coefficient of x^n in the expansion of 1/(n+1) * (1 + x + k*x^2)^(n+1).

%F A(n,k) = Sum_{j=0..floor(n/2)} k^j * binomial(n,j) * binomial(n-j,j)/(j+1) = Sum_{j=0..floor(n/2)} k^j * binomial(n,2*j) * A000108(j).

%F (n+2) * A(n,k) = (2*n+1) * A(n-1,k) - (1-4*k) * (n-1) * A(n-2,k).

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 3, 4, 5, 6, 7, 8, ...

%e 1, 4, 7, 10, 13, 16, 19, 22, ...

%e 1, 9, 21, 37, 57, 81, 109, 141, ...

%e 1, 21, 61, 121, 201, 301, 421, 561, ...

%e 1, 51, 191, 451, 861, 1451, 2251, 3291, ...

%e 1, 127, 603, 1639, 3445, 6231, 10207, 15583, ...

%t T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, 2*j] * CatalanNumber[j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 12 2021 *)

%Y Columns k=0..7 give A000012, A001006, A025235, A025237, A091147, A091148, A091149, A217275.

%Y Main diagonal gives A307906.

%Y Cf. A000108, A107267, A247495, A307855.

%K nonn,tabl

%O 0,9

%A _Seiichi Manyama_, May 06 2019